Chapter 3: Applications of the Derivative

Describe what the second-derivative test is for and how to use it. Sketch graphs and sign charts to illustrate your description.

Restate the Mean Value Theorem so that its conclusion has to do with tangent lines.

Find all roots, local maxima and minima, and inflection points of each function f . In addition, determine whether any local extrema are also global extrema on the domain of f .

$f\left(x\right)=\frac{1}{1+\sqrt{x}}$

Each of the limits in Exercises 7–12 is of the indeterminate form $0·\infty$or $\infty ·0$. Rewrite each limit so that it is (a) in the form $\frac{0}{0}$and then (b) in the form $\frac{\infty }{\infty }$. Then (c) determine which of these indeterminate forms would be easier to work with when applying L’Hopital’s rule.

12.role="math" localid="1648643880542" $\underset{x\to 1}{\lim }\sqrt{x-1}\ln \left(x-1\right)$

If$\underset{x\to c}{\lim }\ln \left(f\right(x\left)\right)=L$, then$\underset{x\to c}{\lim }f\left(x\right)=\_\_\_\_$.

True/False: Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) True or False: If, thenis an inflection point of.

(b) True or False: Ifis concave up on an interval I, then it is positive on I.

(c) True or False: Ifis concave up on an interval I, thenis positive on I.

(d) True or False: Ifdoes not exist andis in the domain of, thenis a critical point of the function.

(e) True or False: Ifhas an inflection point atandis differentiable at, then the derivativehas a local minimum or maximum at.

(f) True or False: Ifand, thenhas a local minimum at.

(g) True or False: The second-derivative test involves checking the sign of the second derivative on each side of every critical point.

(h) True or False: The second-derivative test always produces exactly the same information as the first-derivative test.

If a function f has four critical points, how many calculations after finding derivatives are required in order to apply the first-derivative test? The second-derivative test?

Sketch the graph of a function that satisfies the given description. Label or annotate your graph so that it is clear that it satisfies each part of the description.

A function that satisfies the hypothesis, and therefore the conclusion, of Rolle’s Theorem on $[2,6]$.

Find all roots, local maxima and minima, and inflection points of each function f . In addition, determine whether any local extrema are also global extrema on the domain of f .

$f\left(x\right)=x\ln \left(x\right)$.

Find the error in the following incorrect calculation, and then calculate the limit correctly:

$$\begin{gathered} \underset{x\to 0}{\lim }\frac{x^2-x}{2^x-1} \\ =\underset{x\to 0}{\lim }\frac{2x-1}{\left(\ln 2\right)2^x} \\ =\underset{x\to 0}{\lim }\frac{2}{{\left(\ln 2\right)}^2{2}^x} \\ =\frac{2}{{\left(\ln 2\right)}^2{2}^0} \\ =\frac{2}{{\left(\ln 2\right)}^2} \end{gathered}$$